A popular fruit drink is made by mixing fruit juices. Such a mixture with 25% juice is to be mixed with another mixture that is 5% juice to get 20 liters of a mixture that is 10% juice. How much of each should be used?
Kind of Juice
Number of Liters
Amount of Pure Juice
0.25
x
0.25x
0.05
y
0.1
Let's assume that the amount of the 25% juice mixture to be used is x liters and the amount of the 5% juice mixture to be used is y liters.
The total amount of the mixture is given as 20 liters, so we can set up the equation:
x + y = 20
The amount of pure juice in the 25% juice mixture is 0.25x liters, and the amount of pure juice in the 5% juice mixture is 0.05y liters.
The total amount of pure juice in the mixture is given as 10% of the 20 liters, so we can set up the equation:
0.25x + 0.05y = 0.1(20)
Multiplying both sides of the equation by 100 to eliminate the decimals, we get:
25x + 5y = 10(20)
25x + 5y = 200
Now we have a system of two equations:
x + y = 20
25x + 5y = 200
We can solve this system of equations using substitution, elimination, or matrices. Let's use substitution to find the values of x and y.
From the first equation, we can solve for x:
x = 20 - y
Substituting this value of x into the second equation, we get:
25(20 - y) + 5y = 200
500 - 25y + 5y = 200
-20y = -300
y = 15
Substituting this value of y back into the first equation, we can solve for x:
x + 15 = 20
x = 5
So, we should use 5 liters of the 25% juice mixture and 15 liters of the 5% juice mixture.
Let's assume that the amount of the first mixture (25% juice) is x liters and the amount of the second mixture (5% juice) is y liters.
According to the information given, the mixture should have 20 liters of 10% juice.
We can set up the equation based on the amount of juice:
0.25x + 0.05y = 0.1(20)
Simplifying this equation, we have:
0.25x + 0.05y = 2
Since we have two unknowns (x and y), we need another equation to solve for both variables.
The other equation can be set up based on the amount of liquid:
x + y = 20
Now we have a system of equations:
0.25x + 0.05y = 2
x + y = 20
We can solve this system of equations using the substitution method or elimination method.
Let's use the elimination method in this case. We will multiply the second equation by 0.05 to make the coefficients of y in both equations the same:
0.05(x + y) = 0.05(20)
0.05x + 0.05y = 1
Now we can subtract the second equation from the first equation to eliminate the y variable:
(0.25x + 0.05y) - (0.05x + 0.05y) = 2 - 1
0.25x - 0.05x = 1
0.2x = 1
x = 1 / 0.2
x = 5
Now we can substitute the value of x back into either of the original equations to solve for y. Let's use the second equation:
5 + y = 20
y = 20 - 5
y = 15
Therefore, 5 liters of the mixture with 25% juice and 15 liters of the mixture with 5% juice should be used to obtain 20 liters of a mixture with 10% juice.
To solve this problem, we can set up a system of equations based on the given information.
Let's use x to represent the number of liters of the 25% juice mixture and y to represent the number of liters of the 5% juice mixture.
The total volume of the mixture is given as 20 liters, so we can write the equation:
x + y = 20
The total amount of pure juice in the 25% mixture is 0.25x, and the total amount of pure juice in the 5% mixture is 0.05y. Since we want a mixture that is 10% juice, the equation for the total amount of pure juice is:
0.25x + 0.05y = 0.1(20)
Simplifying the second equation, we get:
0.25x + 0.05y = 2
Now we have a system of equations:
x + y = 20
0.25x + 0.05y = 2
To solve this system, we can use the method of substitution or elimination. Let's use the method of substitution.
Solving the first equation for x, we get:
x = 20 - y
Substituting this value of x into the second equation, we have:
0.25(20 - y) + 0.05y = 2
Expanding and simplifying the equation, we get:
5 - 0.25y + 0.05y = 2
Combining like terms, we have:
-0.2y = -3
Dividing both sides by -0.2, we get:
y = 15
Substituting this value of y back into the first equation, we can find x:
x + 15 = 20
x = 5
Therefore, 5 liters of the 25% juice mixture and 15 liters of the 5% juice mixture should be used to get 20 liters of a 10% juice mixture.